<p> MATHEMATICS PAPER 2 MEMORANDUM VCAA 2016</p><p>Marks: 150 Time: 3 hours</p><p>INSTRUCTIONS</p><p>1. This question paper consists of 28 pages and an Information Sheet of 2 pages. Please check that your paper is complete.</p><p>2. Read the questions carefully.</p><p>3. Answer all questions on the question paper and hand this in at the end of the examination.</p><p>4. Do not work on loose sheets of paper.</p><p>5. Extra space is provided at the end of the paper, should this be necessary.</p><p>6. Diagrams are not necessarily drawn to scale.</p><p>7. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.</p><p>8. Round off to one decimal place, unless specified otherwise.</p><p>9. Ensure that your calculator is in DEGREE mode.</p><p>10. All necessary working details must be clearly shown.</p><p>QUESTION 1 2 3 4 5 6 7 8 9 10 11 12 TOTAL</p><p>MAXIMUM 9 12 20 12 10 12 13 10 14 16 15 7 150</p><p>MARK</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 1 of 25 SECTION A</p><p>QUESTION 1</p><p>(a) Simplify the following expression:</p><p>(5)</p><p>(b) Evaluate the following expression without using a calculator:</p><p>(4)</p><p> or any other correct expansion</p><p>9 marks</p><p>QUESTION 2</p><p>The given data values reflect the masses (in kg) of 20 athletes in the school team:</p><p>40 47 52 53 55 57 57 58 60 x 2 63 64 64 65 66 67 67 68 69 73</p><p>(a) Determine the value of x if the median of the data is 61,5 kg. (2)</p><p>(b) Draw a box-and-whisker diagram of the given data, indicating the necessary values clearly. Use a scale of . (5)</p><p>(c) A value in the data set is considered to be an outlier if that value is either: less than or greater than </p><p>Determine whether 40kg is an outlier of this data set. (3)</p><p> outlier </p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 3 of 25 (d) If 4 kg were added to each data value in the given set, what effect would it have on the resulting (1) mean and (2) standard deviation? (2)</p><p>(1) The mean would increase by 4 kg </p><p>(2) The standard deviation would remain the same </p><p>12 marks QUESTION 3</p><p>(a) , and are given. BA is produced to D and .</p><p>The circle with centre A passes through E, the midpoint of BA.</p><p>4 Determine: (1) the coordinates of E. (2)</p><p>(2) the equation of the circle with centre A. (4)</p><p>(3) the angle of inclination of the line BC. (3)</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 5 of 25 (4) the equation of the line CD. (3)</p><p>(b) The equation of the circle with centre K is given.</p><p>(1) Calculate the coordinates of K and the radius of the circle. (4)</p><p>6 (2) Hence, determine the equation of the tangent at the point on the circle. (4)</p><p>20 marks QUESTION 4</p><p>(a) Refer to the sketch: Q, V, T, S and R lie on the circle. RV and ST are produced to meet at P. PQ is a tangent to the circle at Q. QS is a diameter. </p><p>Complete the table giving the reason for each of the statements. (6)</p><p>STATEMENT REASON</p><p>Tan-chord theorem </p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 7 of 25 Ext cyclic quad </p><p>Tan rad </p><p> in semi-circle OR tan-chord theorem </p><p>Opp cyclic quad </p><p>(b) AB is a tangent to the circle with centre O. BCOD is a straight line and .</p><p>(1) By using the given reason in each case, write a true statement that includes .(3)</p><p>STATEMENT REASON</p><p>a Tan-chord theorem</p><p>a Alt ’s, </p><p>a ’s in the same segment</p><p>(2) Giving reasons, show that . (3)</p><p>STATEMENT REASON</p><p> in semi-circle </p><p>8 Int of </p><p>12 marks QUESTION 5</p><p>(a) Without using a calculator, determine the value of if it is given that and . (5)</p><p>Sketch or Pythag</p><p>(b) If express the following in terms of p without using a calculator: (1) (2)</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 9 of 25 (2) (3)</p><p>(1)</p><p>(2)</p><p>10 marks QUESTION 6</p><p>(a) A, B, C and D are points on a circle with centre O and diameter BD.</p><p>10 and </p><p>Calculate, with reasons, the sizes of the following angles:</p><p>(1)</p><p>(4)</p><p>(2)</p><p>(3)</p><p>STATEMENT REASON</p><p>(1)</p><p> at centre = 2 at circle </p><p>Alternate</p><p> in semi-circle </p><p> at centre = 2 at circle </p><p>(2)</p><p>’s in same segment </p><p>’s opp equal sides </p><p>Alternate</p><p>Int ’s of </p><p>’s opp equal sides (b) In it is given that </p><p> and </p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 11 of 25 Calculate the values of m and k by referring to the lengths on the diagram. (5)</p><p>STATEMENT REASON</p><p>Line one side;</p><p>Line one side;</p><p>12 marks TOTAL SECTION A: 75 SECTION B</p><p>QUESTION 7</p><p>An athlete’s ability to take and use oxygen effectively is called his / her VO2-max. </p><p>12 Twelve athletes with pre-recorded VO2-max readings ran for one hour. The distances (in kilometres) that they each covered are represented in the table:</p><p>VO -max 2 20 55 30 25 40 30 50 40 35 30 50 40 reading Distance run 8 18 13 10 11 12 16 14 13 9 15 12 (km)</p><p>(a) Draw a scatter plot of the data. (Place VO2-max on the horizontal axis.) (4)</p><p>(b) Use the correlation coefficient to describe the correlation between the two sets of data. (2)</p><p>Strong positive linear </p><p>(c) Determine the equation of the least square line of best fit and draw it on the graph. (5)</p><p>) </p><p>On graph </p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 13 of 25 (d) Use the method of interpolation to predict the distance run if an athlete has a VO2-max value of 26. (2)</p><p>On graph </p><p>OR </p><p>13 marks</p><p>QUESTION 8</p><p>(a) Without using a calculator, determine the general solution of:</p><p>(6)</p><p>Alternate</p><p>14 (b) Prove that = 2 (4)</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 15 of 25 10 marks QUESTION 9</p><p>(a) The circle with centre T is given. Calculate the value(s) of m if the length of the tangent from point outside the circle to the point P on the circumference of the circle is . (6)</p><p>16 (b) Refer to the diagram alongside and determine the equation of the circle, with centre M, that touches the x-axis at and passes through the point </p><p>(8)</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 17 of 25 14 marks</p><p>QUESTION 10</p><p>(a) The curves of and are sketched for .</p><p>(1) Write down the values of a, b, c and d. (4)</p><p>(2) Write down the new equation of f if it is translated to the right and 3 units down. (2)</p><p>18 (b) A is the foot of a vertical tower AD. B and C are two points in the same horizontal plane as A. It is given that and , while the angle of elevation of D from B is y°.</p><p>If prove that: (6)</p><p>Alternate:</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 19 of 25 (c) A line is drawn from the origin O to the point .</p><p>The line OP is rotated about the y-axis to form a cone.</p><p> where s is the slant height and r the radius.</p><p>Determine the shaded outer surface area of the resulting cone. (4)</p><p>16 marks QUESTION 11</p><p>20 (a) O is the centre of the circle and The radius of the circle is and .</p><p>Calculate the length of AM (7) STATEMENT REASON</p><p> radii Midpoint Th Pythag Alternate Given in semi-circle is commom AAA Pythag</p><p>Alternate Given in semi-circle</p><p>Pythag Line one side </p><p>(b) KLMN is a trapezium with </p><p> and </p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 21 of 25 Prove:</p><p>(8)</p><p>STATEMENT REASON</p><p> similarity</p><p>Given</p><p>Alt ’s, </p><p>AAA </p><p>Given</p><p>15 marks QUESTION 12</p><p>In the figure P, A, R, C, Q and B are points on the circle.</p><p>Prove that . 22 (7)</p><p>STATEMENT REASON</p><p>Opp ’s cyclic quad </p><p>Opp ’s cyclic quad </p><p>Opp ’s cyclic quad </p><p>Int ’s of </p><p>7 marks TOTAL SECTION B: 75</p><p>VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 23 of 25</p>